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Mortgage Calculations by Hand

First you must define some variables to make it easier to set up: P = principal, the initial amount of the loan I = the annual interest rate (from 1 to 100%) L = length, the length (in years) of the loan, or at least the length over which the loan is amortized.

The following assumes a typical conventional loan where the interest is compounded monthly. First we'll define two more variables to make the calculations easier: J = monthly interest in decimal form = I / (12 x 100) N = number of months over which loan is amortized = L x 12

Now for the big monthly payment (M) formula ... it is:

                              J
         M  =  P  x ------------------------
                 
                      1  - ( 1 + J ) ^ -N

   where 1 is the number one (it does not appear too clearly
   on some browsers)

So to calculate it, you would first calculate 1 + J then take that to the -N (minus N) power, subtract that from the number 1. Now take the inverse of that (if you have a 1/X button on your calculator push that). Then multiply the result times J and then times P.

The one-liner for a program would be (adjust for your favorite language):

         M = P * ( J / (1 - (1 + J) ** -N))

So now you should be able to calculate the monthly payment, M. To calculate the amortization table you need to do some iteration (i.e. a simple loop). Here are the simple steps :

Step 1: Calculate H = P x J, this is your current monthly interest Step 2: Calculate C = M - H, this is your monthly payment minus your monthly interest, so it is the amount of principal you pay for that month Step 3: Calculate Q = P - C, this is the new balance of your principal of your loan. Step 4: Set P equal to Q and go back to Step 1: You thusly loop around until the value Q (and hence P) goes to zero.

Many people have asked how to find N (number of payments) given the payment, interest and loan amount. The answer to the actual formula is in the book: The Vest Pocket Real Estate Advisor by Martin Miles (Prentice Hall). Here's the formula:

n = -1/q * (LN(1-(B/m)*(r/q)))/LN(1+(r/q))

Where:

q = amount of annual payment periods
r = interest rate
B = principle
m = payment amount
n = amount payment periods
LN = natural logarithm

These calculations are only estimates and should not be used to determine actual loan costs. Please consult your tax advisor for information on the deductibility of interest for tax purpose. Refinancing or taking out a home equity loan or line of credit may increase the total number of monthly payments and the total amount paid when comparing to your current situation.

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